36 Integrals with Coalescing Saddles Properties 36.10 Differential Equations 36.12 Uniform Approximation of Integrals

§36.11 Leading-Order Asymptotics

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Keywords:: Pearcey integral, asymptotic approximations, canonical integrals, elliptic umbilic canonical integral, hyperbolic umbilic canonical integral, swallowtail canonical integral
Notes:: The formulas in this section are derived by the method of stationary phase, applied to the real critical points of the integral representations in §36.2. See §2.3(iv) and also Berry and Howls (1991). For (36.11.4) the integral is exponentially small when $x>0$ and the dominant contribution is from a critical point off the real axis.
Referenced by:: §36.15(ii), Erratum (V1.0.19) for Notation
Permalink:: http://dlmf.nist.gov/36.11
See also:: Annotations for Ch.36

With real critical points (36.4.1) ordered so that

36.11.1		$t_{1}(\mathbf{x})<t_{2}(\mathbf{x})<\cdots<t_{j_{\max}}(\mathbf{x}),$
ⓘ Symbols: $t_{j}(\mathbf{x})$ : solutions Permalink: http://dlmf.nist.gov/36.11.E1 Encodings: TeX, pMML, png See also: Annotations for §36.11 and Ch.36

and far from the bifurcation set, the cuspoid canonical integrals are approximated by

36.11.2		$\Psi_{K}\left(\mathbf{x}\right)=\sqrt{2\pi}\sum\limits_{j=1}^{j_{\max}(\mathbf% {x})}\exp\left(i\left(\Phi_{K}\left(t_{j}(\mathbf{x});\mathbf{x}\right)+\tfrac% {1}{4}\pi(-1)^{j+K+1}\right)\right)\left\|\frac{{\partial}^{2}\Phi_{K}\left(t_{% j}(\mathbf{x});\mathbf{x}\right)}{{\partial t}^{2}}\right\|^{-1/2}(1+o\left(1% \right)).$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$ : cuspoid catastrophe of codimension $K$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $t$ : variable, $K$ : codimension and $t_{j}(\mathbf{x})$ : solutions Permalink: http://dlmf.nist.gov/36.11.E2 Encodings: TeX, pMML, png See also: Annotations for §36.11 and Ch.36

Asymptotics along Symmetry Lines

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See also:: Annotations for §36.11 and Ch.36

36.11.3		$\Psi_{2}\left(0,y\right)=\begin{cases}\sqrt{\ifrac{\pi}{y}}\left(\exp\left(% \tfrac{1}{4}i\pi\right)+o\left(1\right)\right),&y\to+\infty,\\ \sqrt{\ifrac{\pi}{\|y\|}}\exp\left(-\tfrac{1}{4}i\pi\right)\left(1+i\sqrt{2}\exp% \left(-\frac{1}{4}iy^{2}\right)+o\left(1\right)\right),&y\to-\infty.\end{cases}$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $y$ : real parameter Permalink: http://dlmf.nist.gov/36.11.E3 Encodings: TeX, pMML, png See also: Annotations for §36.11, §36.11 and Ch.36

36.11.4	$\displaystyle\Psi_{3}\left(x,0,0\right)$	$\displaystyle=\frac{\sqrt{2\pi}}{(5\|x\|^{3})^{1/8}}\begin{cases}\exp\left(-2% \sqrt{2}(\ifrac{x}{5})^{5/4}\right)\left(\cos\left(2\sqrt{2}(\ifrac{x}{5})^{5/% 4}-\tfrac{1}{8}\pi\right)+o\left(1\right)\right),&x\to+\infty,\\ \cos\left(4(\ifrac{\|x\|}{5})^{5/4}-\tfrac{1}{4}\pi\right)+o\left(1\right),&x\to% -\infty.\end{cases}$
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $o\left(\NVar{x}\right)$ : order less than and $x$ : real parameter Referenced by: §36.11 Permalink: http://dlmf.nist.gov/36.11.E4 Encodings: TeX, pMML, png See also: Annotations for §36.11, §36.11 and Ch.36
36.11.5	$\displaystyle\Psi_{3}\left(0,y,0\right)$	$\displaystyle=\overline{\Psi_{3}(0,-y,0)}=\exp\left(\tfrac{1}{4}i\pi\right)% \sqrt{\ifrac{\pi}{y}}\left(1-(i/\sqrt{3})\exp\left(\tfrac{3}{2}i(\ifrac{2y}{5}% )^{5/3}\right)+o\left(1\right)\right),$
$y\to+\infty$ .
ⓘ Symbols: $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $y$ : real parameter Permalink: http://dlmf.nist.gov/36.11.E5 Encodings: TeX, pMML, png See also: Annotations for §36.11, §36.11 and Ch.36
36.11.6	$\displaystyle\Psi_{3}\left(0,0,z\right)$	$\displaystyle=\frac{\Gamma\left(\tfrac{1}{3}\right)}{\|z\|^{1/3}\sqrt{3}}+\begin% {cases}o\left(1\right),&z\to+\infty,\\ \dfrac{2\sqrt{\pi}5^{1/4}}{(3\|z\|)^{3/4}}\left(\cos\left(\dfrac{2}{3}\left(% \dfrac{3\|z\|}{5}\right)^{5/2}-\dfrac{1}{4}\pi\right)+o\left(1\right)\right),&z% \to-\infty.\end{cases}$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $o\left(\NVar{x}\right)$ : order less than and $z$ : real parameter Permalink: http://dlmf.nist.gov/36.11.E6 Encodings: TeX, pMML, png See also: Annotations for §36.11, §36.11 and Ch.36

36.11.7	$\displaystyle\Psi^{(\mathrm{E})}\left(0,0,z\right)$	$\displaystyle=\frac{\pi}{z}\left(i+\sqrt{3}\exp\left(\frac{4}{27}iz^{3}\right)% +o\left(1\right)\right),$
$z\to\pm\infty$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $z$ : real parameter Permalink: http://dlmf.nist.gov/36.11.E7 Encodings: TeX, pMML, png See also: Annotations for §36.11, §36.11 and Ch.36
36.11.8	$\displaystyle\Psi^{(\mathrm{H})}\left(0,0,z\right)$	$\displaystyle=\frac{2\pi}{z}\left(1-\frac{i}{\sqrt{3}}\exp\left(\frac{1}{27}iz% ^{3}\right)+o\left(1\right)\right),$
$z\to\pm\infty$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : exponential function, $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : hyperbolic umbilic canonical integral function, $\mathrm{i}$ : imaginary unit, $o\left(\NVar{x}\right)$ : order less than and $z$ : real parameter Permalink: http://dlmf.nist.gov/36.11.E8 Encodings: TeX, pMML, png See also: Annotations for §36.11, §36.11 and Ch.36

36.10 Differential Equations 36.12 Uniform Approximation of Integrals

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